How do we solve an equation? Solving an equation means that we are determining what value the variable is equal to. That means that we need to get the variable by itself on one side of the equals sign. For example, we want to end up with something that looks like x = 10, y = --2, n = 2r, etc. We do this by using inverse operations to cancel out everything that's on the same side of the equals sign as the variable we're solving for.
Take a look at this example:
Solve the equation for x. (This means that we have to get x by itself on one side of the equals sign)
x + 7 = 12 (Looking at this equation, we can see that the + 7 is on the same side of the equals sign as the x, so that's what we need to get rid of)
- 7 - 7 (To cancel out + 7, we have to subtract 7 because that's the inverse operation. We subtract 7 on both sides of the equals sign)
x = 5 (The + 7 and - 7 cancel each other out, so x is left by itself on the left-hand side of the equation and 12 - 7 = 5)
We know that we are done solving because we have x by itself on one side of the equation and we have a value that x is equal to.
TASK: Solve the following 5 equations on a separate sheet of paper. Show all your work. Turn them in during class.
1. 8 + n = 2
2. r - 12 = -3
3. -2x = 24
4.
5.
In each of the equations above we only had to do one step to get the variable by itself. However, in most equations we will have to do multiple steps. Each step is done the same way as we did the single step equations (using inverse operations to cancel things out that are on the same side of the equation as the variable), but instead of just doing this once, we will have to do it multiple times. For example:
3x - 8 = 13
In this equation, in order to solve for x, we need to cancel out the --8 AND the 3. The question is, which number should we cancel out first? We can cancel either number out first, but it's going to be easier to cancel out the --8 because it is not with the variable. So, we add 8 to both sides of the equation (inverse operations) and we get:
3x = 21
Now, we can cancel out the 3 by dividing both sides of the equation by 3 and we get:
x = 7
Take a look at another example:
-2(x + 4) = 16
In this problem, we can't do anything to get x by itself right away because it's inside the parentheses. Therefore, we have to get rid of the parentheses.
TASK: How do we get rid of parentheses?
Now we'll do the distributive property and we get:
-2x - 8 = 16
Now we have an equation like the one in our previous example. In this equation, we have to cancel out the --8 and the --2. We will cancel out the --8 first because it's not with the variable. We'll cancel it by adding 8 to both sides of the equation. We get:
-2x = 24
Now we divide by --2 on both sides (we don't add two because the operation being done in the equation is multiplication, not subtraction). We get:
x = -12
TASK: Write at least 5 sentences explaining how we solve equations. Be specific. Your paragraph should answer the following questions: How do we decide which operation to do when solving equations? How do we decide what to do first when solving equations?
TASK: Solve the following 5 equations on a separate sheet of paper. Show all your work. Turn in during class.
How do we solve an equation? Solving an equation means that we are determining what value the variable is equal to. That means that we need to get the variable by itself on one side of the equals sign. For example, we want to end up with something that looks like x = 10, y = --2, n = 2r, etc. We do this by using inverse operations to cancel out everything that's on the same side of the equals sign as the variable we're solving for.
Take a look at this example:
Solve the equation for x. (This means that we have to get x by itself on one side of the equals sign)
x + 7 = 12 (Looking at this equation, we can see that the + 7 is on the same side of the equals sign as the x, so that's what we need to get rid of)
- 7 - 7 (To cancel out + 7, we have to subtract 7 because that's the inverse operation. We subtract 7 on both sides of the equals sign)
x = 5 (The + 7 and - 7 cancel each other out, so x is left by itself on the left-hand side of the equation and 12 - 7 = 5)
We know that we are done solving because we have x by itself on one side of the equation and we have a value that x is equal to.
TASK: Solve the following 5 equations on a separate sheet of paper. Show all your work. Turn them in during class.
1. 8 + n = 2
2. r - 12 = -3
3. -2x = 24
4.
5.
In each of the equations above we only had to do one step to get the variable by itself. However, in most equations we will have to do multiple steps. Each step is done the same way as we did the single step equations (using inverse operations to cancel things out that are on the same side of the equation as the variable), but instead of just doing this once, we will have to do it multiple times. For example:
3x - 8 = 13
In this equation, in order to solve for x, we need to cancel out the --8 AND the 3. The question is, which number should we cancel out first? We can cancel either number out first, but it's going to be easier to cancel out the --8 because it is not with the variable. So, we add 8 to both sides of the equation (inverse operations) and we get:
3x = 21
Now, we can cancel out the 3 by dividing both sides of the equation by 3 and we get:
x = 7
Take a look at another example:
-2(x + 4) = 16
In this problem, we can't do anything to get x by itself right away because it's inside the parentheses. Therefore, we have to get rid of the parentheses.
TASK: How do we get rid of parentheses?
Now we'll do the distributive property and we get:
-2x - 8 = 16
Now we have an equation like the one in our previous example. In this equation, we have to cancel out the --8 and the --2. We will cancel out the --8 first because it's not with the variable. We'll cancel it by adding 8 to both sides of the equation. We get:
-2x = 24
Now we divide by --2 on both sides (we don't add two because the operation being done in the equation is multiplication, not subtraction). We get:
x = -12
TASK: Write at least 5 sentences explaining how we solve equations. Be specific. Your paragraph should answer the following questions: How do we decide which operation to do when solving equations? How do we decide what to do first when solving equations?
TASK: Solve the following 5 equations on a separate sheet of paper. Show all your work. Turn in during class.
1.
2.
3.
4.
5.